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Gaussian process approximations for multicolor Pólya urn models

Published online by Cambridge University Press:  25 February 2021

Konstantin Borovkov*
Affiliation:
The University of Melbourne
*
*Postal address: School of Mathematics and Statistics, The University of Melbourne, Parkville3010, Australia. Email address: borovkov@unimelb.edu.au

Abstract

Motivated by mathematical tissue growth modelling, we consider the problem of approximating the dynamics of multicolor Pólya urn processes that start with large numbers of balls of different colors and run for a long time. Using strong approximation theorems for empirical and quantile processes, we establish Gaussian process approximations for the Pólya urn processes. The approximating processes are sums of a multivariate Brownian motion process and an independent linear drift with a random Gaussian coefficient. The dominating term between the two depends on the ratio of the number of time steps n to the initial number of balls N in the urn. We also establish an upper bound of the form $c(n^{-1/2}+N^{-1/2})$ for the maximum deviation over the class of convex Borel sets of the step-n urn composition distribution from the approximating normal law.

Type
Research Papers
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Applied Probability Trust

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